A swimmer crossing a river proceeds at an absolute speed of

\"A swimmer crossing a river proceeds at an absolute speed of 2.5 m/s on a course oreinted at a 45^o angle to the 1 m/s current. Given that the absolute velocity of the swimmer is equal to the vector sum of the velocity of the current and the velocity of the swimmer with respect to the current, what is the magniturde and direction of the velocity of the swimmer with respect to the current?\" (Answer: 3.3 m/s at an angle of 32.6^o to the current)

I am completely lost on this question and honestly do not even know where to start. If this question could be broken down for me that would be great.

Solution

Here, we have a swimmer who has directed himself at 45 degrees to the stream which is flowing at the speed of 1 m/s. The swimmer\'s speed of 2.5 m/s will have two components, one along the stream and the other perpendicular to it.

Now, the swimmer will get an extra speed down the current because of the current\'s speed, while its own speed along the perpendicular to the current will remain the same.

So we can say that the swimmer\'s net speed down the current would be: Current speed + Downward component of swimmer\'s speed.

That is, Vs = 1 + 2.5Cos45 = 2.768 m/s

Also, the swimmer\'s speed along the perpendicular would remain same, hence Vp = 2.5 cos45 = 1.768 m/s

Therefore the net speed of the swimmer would be vector sum of these two velocities,

That is, Vnet = sqrt[Vs² + Vp²]

or, Vnet = 3.284 m/s = 3.3 m/s

Also, tangent of angle it makes with the current = Vp / Vs

or, Angle with the current = 32.56 degrees

NOTE: You need to consider it this way: The swimmer \'tries\' to flow at an angle of 45 degrees to the current, but the current gives him some extra speed downstream. However, his speed\'s component along the perpendicular to the stream will remain the same as he will suffer no extra \'push\'

Imagine yourself not swimming at all. What happens then? You get a speed equal to the stream\'s speed downstream, while you get no speed in the perpendicular direction. You just need to replace it with the situation in which you are trying to flow at some speed at an angle to the stream.